I would like to know whether the following expressions are correct. I am unsure about the usage "would + have to" and "might + have to". If such expressions exist, kindly tell me the meaning of those expressions.

Those are all valid sentences. (3) and (4) express future possibility. (1) and (2) need more context to explain the use of "would". There could be an implied condition ("if"). They could also express past habitual behaviour (a more formal usage mostly confined to written English).

I have to take a taxi. (I must take a taxi. I have no choice.)I might have to take a taxi. (Maybe I will have to take a taxi.)I would have to take a taxi. (if certain conditions apply)

In the case of would, the speaker might be thinking of a situation where the person who was supposed to pick him up at the airport was unable to do so at the last minute. In that case he would have to take a taxi.